Questions tagged [kerr-metric]

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Effective potential for Kerr geometry

In the review Foundations of Black Hole Accretion Disk Theory, the authors defines an effective potential for Kerr geometry as (Chap. 2, eqn. 23) $$\mathcal{U}_{eff}=-\frac{1}{2}\ln\left|g^{tt}-2lg^{t\...
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What is exactly rotating in a rotating black hole?

I have read this: https://arxiv.org/abs/gr-qc/9404041 In General Relativity the black hole solutions which have so far been found form a four parameter family called the generalized Kerr-...
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How to derive the angular velocity of circular orbits in Kerr geometry?

I am trying to derive the angular velocity of a circular orbit in Kerr geometry, eqn.(2.16) in Bardeen et al (1972) which reads $$\Omega=\dfrac{1}{r^{3/2}+a}$$ (Note that I am using the units in which ...
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Derivation of radial momentum equation in Kerr geometry

I am trying to derive the radial momentum equation in the equatorial plane of Kerr geometry obtained by Lasota (1994) which reads (eqn. 6 in page-343; I am using units in which $M=1$) as follows: $$uu'...
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Physical significance of angular velocity of orbits around Kerr black holes

For the Kerr metric $$ds^2=\left(g_{tt}-\frac{g_{t\phi}^2}{g_{\phi\phi}}\right)dt^2+g_{\phi\phi}\left(d\phi-\omega dt\right)^2+g_{rr}dr^2+g_{\theta\theta}d\theta^2$$ the angular momentum is defined as ...
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Zero mass Kerr metric

When mass in Kerr metric is put to zero we have $$ds^{2}=-dt^{2}+\frac{r^{2}+a^{2}\cos^{2}\theta}{r^{2}+a^{2}}dr^{2}+\left(r^{2}+a^{2}\cos^{2}\theta\right)d\theta^{2}+\left(r^{2}+a^{2}\right)\sin^{2}\...
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Metric for a rotating star

If we want to describe a static spherically symmetric star we can use a metric which matches the Schwarzschild solution with correct mass on the outside of the star but differs from Schwartzschild in ...
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Light-like normal vectors

Can someone please show me how to mathematically establish that the normal vector to the event horizon of a Kerr Black Hole is light-like?
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143 views

Numerical Solutions for Equatorial Orbits in the Kerr Black Hole

Currently, I am trying to find timelike orbits in the Kerr metric around the equator. The problem is that no matter which parameters I choose or the method I use I can't seem to get to physically ...
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How to compute Kerr geodesics?

How would I start to numerically compute trajectories of Kerr geodesics with constants of motion like in this wikipedia page. I want to recreate trajectories like in this picture in Matlab.
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How does the Penrose diagram for a spinning black hole differ in realistic scenarios (formed by stellar collapse)?

The Penrose diagram for a non-spinning Schwarzschild black hole is Notably, there is a second universe "on the other side" of the black hole. However, actual black holes form by stellar collapse, and ...
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Metric diameter of a ring singularity

In the Kerr metric the ring singularity is located at the coordinate radius $r=0$, which corresponds to a ring with the cartesian radius $R=a$. So the center of the ring singularity in cartesian ...
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1answer
85 views

Quadrupole moment of Kerr spacetime

In this paper this paper, the Kerr black hole is described as having quadrupole moment of $q=J^2/M$ (which means $q=a^2M$ using $J=aM$) whereas in this paper it says in the abstract that the limiting ...
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For a given mass, how big can a Kerr black hole get?

We know that in a Kerr Black Hole the singularity is in the form of a 1 dimensional ring. If we have a 25 solar mass black hole, how big would the Kerr Ring be, width wise? Also, I read the Wiki on ...
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Can anyone tell me how can draw shadow of black hole like in presented in Intersteller movie? Is there any code for it in Mathematica or in Python? [closed]

Equation of motion for photon $$ \Sigma \frac{dt}{d\lambda} = aL\left(1-\frac{r^2+a^2}{\Delta}\right) + \omega\left(\frac{\left(r^2+a^2\right)^2}{\Delta}-a^2 \sin ^2\theta\right)\ , $$ $$ \Sigma\frac{...
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Gravitational lensing redshift around a Kerr black hole

Light from a source passes by a Kerr black hole on two sides at the equator and converges at the observer. The axis of rotation of the black hole is perpendicular to the direction of light. Two rays ...
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Beyond Kerr Carter constant?

What are the most symmetrical black hole spacetimes whose motion is completely integrable with a Carter constant-like and hidden symmetry superintegrability condition? Do type D-spacetimes have a ...
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Is a stable orbit possible inside the ergosphere of a Kerr (spinning) black hole?

I have heard that it's "impossible to hover" inside of an ergosphere, but everywhere I read this seemed to be speaking in the context of "relative to a stationary observer outside of the ergosphere". ...
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4answers
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Visualization of $ dtdx$ and $dxdy$ term in metric tensor

For the sake of simplicity, lets take a 2+1 dimensional spacetime. Lets take the metric $$ds^2 = g_{tt}dt^2 + g_{xx}dx^2 + g_{yy}dy^2 + g_{tx}dtdx + g_{xy}dxdy$$ What is the visualization or ...
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Centrifugal force on spinning black hole?

I saw the term spinning black hole popping up everywhere so my question do spinning black hole behave similarly to say a planet where it bulge in the equatorial and compress at the poles? what ...
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Kerr Black hole EH and Ergosphere embedding

Goodmorning everyone. I would like to share with you a question that has been gripping me for some time, but which I have never been able to give a convincing answer. When representing the ergosphere ...
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1answer
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What would happen to the Earth, if the moon was a black hole? [closed]

Would it be a feasible scenario? I have read this question: What would happen to the Moon if Earth is turned into a black hole? Where Lubos Motl says: The extremal Kerr J=GM2/c∼RbhMc. Now, the ...
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Periodicity trick for Kerr Black Holes

I am slightly confused concerning the euclidean section of a Kerr black hole. In page 5 of the following paper https://arxiv.org/abs/hep-th/9908022 it is said that in order to get the euclidean ...
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Proof that the Kerr metric may be written in orthogonal form

Prove or disprove that the Kerr metric can be expressed in a set of orthogonal coordinates over some coordinate chart. Motivation for this question stems from my understanding that a metric can ...
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Kerr metric in BMS (Bondi-Metzner-Sachs) coordinates

I am trying to put the Kerr metric into the famous Bondi gauge, which is given for instance by the formula (6.2.10) at page 154 of the following paper: https://arxiv.org/abs/1801.01714. Now, Barnich ...
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Derivation of Equation of Trajectory around a Kerr Black Hole

I was trying to derive equation of motion for test particle around a Kerr black hole. My work is as follows: The Kerr metric is as follows $$ \mathrm ds^2 = -\left(1-\dfrac{2Mr}{\rho^2}\right)\...
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1answer
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Particle is not at rest with respect to itself around Kerr spacetime?

Assuming a particle is at rest in a certain frame of reference around a kerr spacetime and hence the 4-velocity (in spherical polar coordinates) of the particle is given by $$ u=(u^{t},0,0,0)$$ Now, ...
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Can centrifugal force overcome other forces (in a singularity/Kerr metric)?

I have read these questions and answers. None of them answered my question. Typical rotation speeds for black holes Is there a physical upper limit on how fast a physical object can rotate? ...
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1answer
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Complex-valued event horizon of a Kerr black hole?

The Kerr metric has two physical relevant surfaces on which it appears to be singular. Solving the quadratic equation $1/g_{rr} = 0$ yields the solution: $$r_H^\pm=\frac{G M}{c^2}\pm\sqrt{\left(\frac{...
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Would naked singularities have a complex irreducible mass?

The formula for the irreducible mass, also known as the Christodoulou and Ruffini equation, is $$M_{\rm irr} = \frac{\sqrt{2 M^2-Q^2+2 M \sqrt{M^2-Q^2-a^2}}}{2}$$ where M is the mass equivalent of ...
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Simulating a Test Particle in a Kerr Spacetime $(M,\mathcal{O}, \mathcal{A},\nabla^{L.C.})$

The equations of motion for a test particle in a Kerr spacetime $(M,\mathcal{O}, \mathcal{A},\nabla^{L.C.})$ are dictated by four degrees of freedom (i.e. invariant mass $m$ in $p^\mu g_{\mu\nu}p^{\nu}...
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Is a Kerr ring singularity a closed string?

I’m obviously not a scientist, but has anyone considered that a Kerr ring singularity might basically be a closed string? The singularity spins in one direction only and is incredibly flat and thin (...
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What is the escape velocity in the Kerr metric? [duplicate]

In the non-rotating Schwarzschild metric there is a straightforward expression for the escape velocity in the radial direction, defined from the point of view of a stationary observer at that radius, ...
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Physical reality of inner event horizon and inner ergosurface in a rotating black hole in D. Wiltshire et al. “The Kerr spacetime”

In chapter 1/The Kerr spacetime-a brief introduction by Matt Visser of D. Wiltshire, M. Visser, S.M. Scott "The Kerr spacetime - Rotating black holes in general relativity" the author presents a ...
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What is the true shape of a black hole?

I’m pretty sure if a black hole is rotating, then it is shaped as an oblate spheroid. But I know that the equations of general relativity tell that instead of having just one radius. In location of ...
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259 views

The Killing vector $\chi=\partial_t+\Omega_H\partial_\phi$ doesn't look normal to the Killing horizon for a Kerr BH

As mentioned in Carroll's Spacetime and Geometry p. 244, a Killing vector is normal to its Killing horizon. With some help from the other forum, I could check this is true. (FYI, here the Killing ...
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Particle crossing the outer event horizon of a Kerr black hole

I am quite puzzled by the following statement in Sean Carroll's 'Spacetime and geometry' (formula 6.100). A particle with momentum $p^\mu$ crossing the outer event horizon of a Kerr black hole $r=...
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Kerr BH effects on inspiral binaries

In the gravitational wave calculation for binary systems: what is the effect of rotation of two BH (or neutron stars, BH-NS,...) on the usual calculations? Is there any EXACT result known?
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Calculate Komar mass and Komar angular momentum for the Kerr metric [closed]

I have this questions for HW: calculate Komar mass and Komar angular momentum for the Kerr metric. the quations that I see in the lecture notes are: in the notes it dosent explain the parameters ...
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1answer
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Moment of inertia of a rotating black hole [duplicate]

The moment of inertia of a massive object about a given axis describes how its mass is distributed about that axis. I understand that a rotating black hole of a given mass and angular velocity ...
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Can rotating black hole have toroidal event horizon with penetrating relativistic jet(s)?

It is conjectured that a rotating black hole has at its center a ring-shaped singularity. Thus, at the center of the ring-shaped singularity the gravitational field must be zero (similar to ...
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If a black hole is made by a hypothetical particle above Planck mass and it has angular momentum will it be a ring Kerr black hole? [duplicate]

If a single subatomic hypothetical fundamental particle incapable of decay (such as an electron) with a mass exceeding Planck mass possessed angular momentum and it collapsed into a black hole, would ...
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de-Sitter spacetimes

In classical textbooks for GR, Schwarzschild and Kerr spacetimes are adequately described. In which books or articles, it is mostly believed that Reissner-Nordstrom, Kerr-Newman, Schwarzschild-de ...
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Area of the event horizon of a rotating black hole

The Kerr metric for a black hole of mass $M$ and angular momentum $J = aM$ is $$ds^{2} = - \frac{\Delta(r)}{\rho^{2}}(dt-a\sin^{2}\theta d\phi)^{2} + \frac{\rho^{2}}{\Delta(r)}dr^{2} + \rho^{2} d\...
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Escape velocity from a rotating black hole

Under Newton, the escape velocity is $$v_{esc} = \rm c \ \sqrt{r_s/r}$$ where $\rm r_s=2 \ GM/c^2$. In the nonrotating relativistic case (the Schwarzschild case) the radial escape velocity is the ...
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How to get the black hole spin in RPM if you have the value of $a$?

Most research papers use the variable "a" to describe black hole spin, which always has a value of less than 1, I don't know what this variable is called, but from what I know it can't exceed 1 ...
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Conservation equation of a perfect fluid in Kerr geometry

I am reading a paper in which a perfect fluid in the Kerr geometry is studied. The stress-energy tensor is \begin{equation} T_{\mu \nu} = (\epsilon + P) u_\mu u_\nu + P g_{\mu \nu}, \end{equation} ...
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Gram-Schmidt Orthonormalization

In the paper from Kulkarni et al., 2011 (Appendix B1), a method is given for transforming from Boyer-Lindquist (BL) coordinates to a comoving frame. This involves using Gram-Schmidt ...
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Is there a Birkhoff-like theorem for stationary axisymmetric metrics?

I know about the theorem by Robinson and Carter about the uniqueness of the Kerr metric in the case of stationary axisymmetric (SA) black holes. Are there any uniqueness theorems like Birkhoff's ...
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Killing tensor in the Kerr metric

It was famously shown by Carter that the Kerr metric possesses a 4th non-obvious constant of the motion, derived from the separability of the Hamiltonian. This constant is related to a Killing tensor. ...